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Can anyone give a hint as to how to solve a differential equation of the form $$(y')^2=\sum_{i=0}^3 c_iy^i$$ where $y=y(x)$ and $c_i$ are constants? I don't know if this is relevant but it is given that $y'(x),y''(x)\to 0$ as $|x|\to \infty$.

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up vote 4 down vote accepted

Taking the square root and dividing by the right-hand side yields and integral like this one, an elliptic integral that can't be solved with elementary functions for general $c_i$.

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